Overview: The investigator proposes a program of research, education, and outreach on problems in dynamics. The research entails problems in ergodic theory, studying questions of convergence, recurrence, and equidistribution, and problems in topological dynamics, studying properties of recurrence, shift systems, and complexity. The motivation for much of the research is combinatorial in nature, and many of the proposed methods are of a combinatorial flavor. The research goals are a deeper understanding of the connections among these fields and the outreach goals are to broaden the cohort of researchers working in these areas. Intellectual Merit: A basic problem in this area it to understand the long term behavior of systems whose dynamics is too complicated or too chaotic to be understood locally. In symbolic and topological dynamics, the PI proposes building on past results to further understand relations between complexity, periodicity, dynamical properties of the system, and the algebraic properties of naturally associated groups. In ergodic theory, the PI proposes building on previous research in multiple recurrence and convergence, with the study of different averages, equidistribution results, and building of systems with particular properties. The problems proposed lie within dynamics, but there are strong connections to problems from other fields, including combinatorics, number theory, combinatorics of words, and computer science. The PI will continue to explore these deep links, developing applications to these other areas and making use of recent advances in these other areas to address problems within dynamics. Broader Impacts: The outreach portion of the project involves conference organization, mentoring, advising, and public speaking. The PI is actively involved in education of researchers in fields with links to dynamics, organizing interdisciplinary meetings, and lecturing for researchers in related fields, and in education of the general public, giving expository lectures aimed at non-mathematicians. The PI will continue working with the graduate and undergraduate programs at Northwestern University, directing undergraduates in advanced topics and graduate students in doctoral research, running a mentoring group for women at all levels within the Mathematics, Statistics, and Applied Mathematics Departments at Northwestern University, and running a large annual conference for undergraduate women from across the country, designed to increase the number of women applying to graduate school in mathematics.
|Effective start/end date||6/1/18 → 5/31/22|
- National Science Foundation (DMS-1800544)
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